Given:
- f(x+4) - f(x+2) + f(x) = 0

To find:
- The period of the function f(x)

Solution:
To determine the period of the function, we need to find a value T such that f(x+T) = f(x) for all real values of x.

Step 1:
Let's consider a new variable y = x + 4. This implies x = y - 4.

Substituting the value of x in the given equation, we get:
f(y) - f(y - 2) + f(y - 4) = 0

Step 2:
Now, let's consider another new variable z = y - 2. This implies y = z + 2.

Substituting the value of y in the equation obtained in step 1, we get:
f(z + 2) - f(z) + f(z - 2) = 0

Step 3:
Now, if we compare the equations obtained in step 1 and step 2, we can observe that:
f(z) = f(z - 2) - f(z + 2)

Step 4:
Let's consider a new variable w = z - 2. This implies z = w + 2.

Substituting the value of z in the equation obtained in step 3, we get:
f(w) = f(w - 2) - f(w + 2)

Step 5:
If we compare the equations obtained in step 3 and step 4, we can observe that:
f(w) = f(w - 2) - f(w + 2) = f(z) = f(z - 2) - f(z + 2)

Step 6:
Now, let's consider a new variable v = w + 2. This implies w = v - 2.

Substituting the value of w in the equation obtained in step 5, we get:
f(v - 2) = f(v - 4) - f(v)

Step 7:
Comparing the equations obtained in step 5 and step 6, we can observe that:
f(v - 2) = f(v - 4) - f(v) = f(w) = f(w - 2) - f(w + 2)

Step 8:
Considering a new variable u = v - 4. This implies v = u + 4.

Substituting the value of v in the equation obtained in step 7, we get:
f(u) = f(u - 2) - f(u + 4)

Step 9:
Finally, let's compare the equations obtained in step 7 and step 8:
f(u) = f(u - 2) - f(u + 4) = f(v - 2) = f(v - 4) - f(v)

Conclusion:
From step 9, we can observe that f(u) = f(u - 2) - f(u + 4) = f(v - 2) = f(v - 4) - f(v) = f